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Tilted nonparametric regression function estimation

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posted on 2022-10-20, 21:58 authored by Farzaneh Boroumand

In recent decades, nonparametric regression models have gained considerable attention in theoretical and applied statistics because they do not rely on a predetermined functional form of relationship between predictors and the outcome. In this thesis, we study a class of nonparametric regression estimator called linear smoother. We apply a tilting technique to improve the performance of the linear smoother by minimizing the distance between the linear smoother and a comparator. The tilted linear smoother takes advantage of both linear smoother and the comparator. We theorize the convergence rate of the tilted version of linear smoother, which is obtained by minimizing the distance to an infinite order flat-top trapezoidal kernel estimator (IO). We prove that the proposed estimator achieves a high level of accuracy. Moreover, it preserves the attractive properties of the infinite order flat-top kernel estimator. We also present an extensive numerical study for analyzing the performance of two members of this class of estimators, named tilted Nadaraya-Watson (NW) and tilted local linear (LL), for finite samples. The simulation study shows that under some conditions the tilted estimators (tilted NW and tilted LL), are superior to conventional estimators (NW and LL) while under all conditions they perform better than the comparator, IO. We also extended the methodology for multiple predictors in an additive setting which, has led to the development of the Tilted Additive model (TAM). The TAM can be formulated by replacing the standard linear model combination of predictors, such as ∑βjXj with ∑fj(Xj), where fj is a smooth nonlinear function of jth predictor Xj ,respectively, which are estimated from the data. We use a tilted linear smoother to estimate fj , which leads to the development of the TAM. Finally, the performance of the tilted linear smoothers is illustrated by curve fitting to COVID-19 data (daily new cases and daily deaths) for 12 countries. Moreover, we used the TAM to detect significant predictors on the area occupied by the vessels (AOV) in the ocular surface in diabetes mellitus (DM) patients. As DM can affect the microvasculature, we evaluate the effect of different clinical parameters on the vascular density of ocular surface microvasculature in diabetic patients. In this cross-sectional study, red-free conjunctival photographs of diabetic individuals aged 30-60 were taken under defined conditions and analyzed using a Radon transform-based algorithm for vascular segmentation. The AOV images of different diameters were calculated to establish the sum of AOV of different sized vessels. We use a tilted nonparametric regression estimator to estimate the nonlinear effect of predictors on the outcome in the additive setting. The results show Age (p-value=0.019) and Mean Arterial Pressure (MAP) have a significant linear effect on AOV (p-value=0.034). We also find a nonlinear association between Body Mass Index (BMI), daily Urinary Protein Excretion (UPE), Hemoglobin A1C, and Blood Urea Nitrogen (BUN) with AOV. As many predictors do not have a linear relationship with the outcome, we conclude that the TAM will help better elucidate the effect of the different predictors.


Table of Contents

1 Introduction -- 2 Nonparametric Curve Estimation -- 3 Tilted Nonparametric Regression Estimation -- 4 Tilted Additive Model -- 5 Real Data Applications -- 6 Conclusion -- A Appendix A: Cross-Validation Function Plots -- B Appendix B: R code -- References


A thesis submitted to Macquarie University for the degree of Doctor of Philosophy Faculty of Science and Engineering Department of Mathematics and Statistics

Awarding Institution

Macquarie University

Degree Type

Thesis PhD


Thesis (PhD), Department of Mathematics and Statistics, Faculty of Science and Engineering, Macquarie University

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Hassan Doosti


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